RUS  ENG
Full version
JOURNALS // Dal'nevostochnyi Matematicheskii Zhurnal

Dal'nevost. Mat. Zh., 2005, Volume 6, Number 1-2, Pages 117–145 (Mi dvmg205)

Control problems for stationary models of magnetic hydrodynamics of a viscous incompressible fluid
G. V. Alekseev

References

1. A. G. Kulikovskii, G. A. Lyubimov, Magnitnaya gidrodinamika, Fizmatgiz, M., 1962, 248 pp.
2. Dzh. Sherklif, Kurs magnitnoi gidrodinamiki, Mir, M., 1967, 320 pp.
3. L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika, v. 8, Elektrodinamika sploshnykh sred, Nauka, M., 1982, 624 pp.  mathscinet
4. V. V. Nikolskii, Elektrodinamika i rasprostranenie voln, Nauka, M., 1978, 544 pp.
5. O. A. Ladyzhenskaya, V. A. Solonnikov, “O razreshimosti nestatsionarnykh zadach v magnitnoi gidrodinamike”, Dokl. AN SSSR, 124 (1959), 26–28  mathscinet  zmath
6. O. A. Ladyzhenskaya, V. A. Solonnikov, “Reshenie nekotorykh nestatsionarnykh zadach magnitnoi gidrodinamiki dlya vyazkoi neszhimaemoi zhidkosti”, Tr. Matem. in-ta im. V. A. Steklova, 59, 1960, 115–174  mathnet
7. V. A. Solonnikov, “O nekotorykh statsionarnykh kraevykh zadachakh magnitnoi gidrodinamiki”, Tr. Matem. in-ta im. V. A. Steklova, 59, 1960, 174–187  mathnet  mathscinet
8. R. H. Dyer, D. E. Edmunds, “A uniqueness theorem in magnetohydrodynamics”, Arch. Rat. Mech. Anal., 8 (1961), 254–262  crossref  mathscinet  zmath
9. R. H. Dyer, D. E. Edmunds, “On the existence of solutions of the equations of magnetohydro-dynamics”, Arch. Rat. Mech. Anal., 9 (1962), 403–410  crossref  mathscinet  zmath
10. D. E. Edmunds, “Sur I'unicité des solutions des équations de la magnétohydrodynamique”, C.r. Acad. Sci. Paris, 254 (1962), 1377–1379  mathscinet  zmath
11. D. E. Edmunds, “Sur les équations différentielles de la magnétohydrodynamique”, C.r. Acad. Sci. Paris, 254 (1962), 4248–4250  mathscinet
12. J. Förste, “Ein Existenzsatz für stationäre Strömungen in der Magnetohydrodynamik”, Mon. Deutsch. Wiss. Berlin, 6 (1964), 886–894  mathscinet  zmath
13. J. Förste, “Ein Einzigkeitssatz für stationäre Strömungen in der Magnetohydrodynamik”, Mon. Deutsch. Wiss. Berlin, 9 (1967), 241–247  zmath
14. G. Lassner, “Über ein Rand-Anfangswertproblem der Magnetohydrodynamik”, Arch. Rat. Mech. Anal., 30 (1967), 388–405  crossref  mathscinet
15. E. Sanchez-Palencia, “Existence des solutions de certains problémes aux limites en magnetohydrodynamique”, J. Méc, 7:3 (1968), 405–426  mathscinet  zmath
16. E. Sanchez-Palencia, “Quelques résultats d'existence et d'unicité pour des écoulements magnétohydrodynamique non stationnaires”, J. Méc, 8:4 (1969), 509–541  mathscinet  zmath
17. G. V. Alekseev, “O suschestvovanii techeniya provodyaschei zhidkosti v slabo iskrivlennom kanale”, Dinamika sploshnoi sredy, 3, Izd-vo IG SO RAN, Novosibirsk, 1969, 7–16
18. G. G. Branover, A. B. Tsinober, Magnitnaya gidrodinamika neszhimaemykh sred, Nauka, M., 1970, 380 pp.
19. G. Duvaut, J.-L. Lions, “Inéquations en thermoélasticité et magnétohydrodynamique”, Arch. Rat. Mech. Anal., 46 (1972), 241–279  crossref  mathscinet  zmath
20. Sh. Sakhaev, V. A. Solonnikov, “Otsenki resheniya odnoi kraevoi zadachi magnitnoi gidrodinamiki”, Tr. Matem. in-ta im. V. A. Steklova, 127, 1975, 87–108  mathnet  zmath
21. L. I. Stupyalis, “Nestatsionarnaya zadacha magnitnoi gidrodinamiki”, Zap. nauch. seminarov LOMI, 52, 1975, 175–217  mathnet  zmath
22. L. I. Stupyalis, “O razreshimosti nachalno-kraevoi zadachi magnitnoi gidrodinamiki”, Zap. nauch. seminarov LOMI, 69, 1977, 219–239  mathnet  zmath
23. O. A. Ladyzhenskaya, V. A. Solonnikov, “The linearization principle and invariant manifold problems of magnetohydrodynamics”, J. Sov. Math., 8 (1977), 384–422  crossref  zmath
24. J. Förste, “Über die Grundgleichungen der Plasmadynamick auf der Basis der Zweiflüssigke- itstheorie”, Z. Angew. Math. Mech., 59 (1979), 553–558  crossref  mathscinet  zmath
25. L. I. Stupyalis, “Nestatsionarnaya zadacha magnitnoi gidrodinamiki dlya sluchaya dvukh prostranstvennykh peremennykh”, Kraevye zadachi matematicheskoi fiziki, Tr. Matem. in-ta im. V. A. Steklova, 147, no. 10, 1980, 156–168  mathnet  zmath
26. L. I. Stupyalis, “Ob odnoi kraevoi zadache dlya statsionarnoi sistemy uravnenii magnitnoi gidrodinamiki”, Kraevye zadachi matematicheskoi fiziki, Tr. Matem. in-ta imeni V. A. Steklova, 147, no. 10, 1980, 169–193  mathnet  zmath
27. G. V. Alekseev, “O razreshimosti odnorodnoi kraevoi zadachi dlya uravnenii magnitnoi gidrodinamiki idealnoi zhidkosti”, Dinamika sploshnoi sredy, 57, Izd-vo IG SO RAN, Novosibirsk, 1982, 6–24
28. M. Sermange, R. Temam, “Some mathematical questions related to the MHD equations”, Comm. Pure. Appl. Math., 36 (1983), 635–664  crossref  mathscinet  zmath  adsnasa  isi
29. Z. Yoshida, Y. Giga, “On the Ohm-Navier-Stokes system in magnetohydrodynamics”, J. Math. Phys., 1983, 2860–2864  crossref  mathscinet  zmath  adsnasa  isi
30. S. V. Chizhonkov, “Ob odnoi sisteme uravnenii tipa magnitnoi gidrodinamiki”, Dokl. AN SSSR, 278:5 (1984), 1074–1077  mathnet  mathscinet  zmath  adsnasa
31. Y. Giga, Z. Yoshida, “On the equations of the two-component theory in magnetohydro-dynamics”, Communs. Partial Diff. Eqns., 9 (1984), 503–522  crossref  mathscinet  zmath
32. D. Ebel, M. C. Shen, “On the linear stability of a toroidal plasma with resistivity, viscosity, and Hall effect”, J. Math. Anal. Appl., 125 (1987), 81–103  crossref  mathscinet  zmath  isi
33. D. Ebel, M. C. Shen, “Linearization principle for a toroidal Hall current plasma with viscosity and resistivity”, Anal. Mat. Pura Appl., 150 (1988), 39–65  crossref  mathscinet  zmath
34. J. Blum, Numerical simulation and optimal control in plasma physics: with applications in tokamaks, Gauthier-Villars, Paris, 1989  mathscinet  zmath
35. Y. Giga, Z. Yoshida, “A dynamic free-boundary problem in physics”, SIAM J. Math. Anal., 21:5 (1990), 1118–1138  crossref  mathscinet  zmath  isi
36. V. L. Pospelov, “Ob ustoichivosti statsionarnogo resheniya odnoi zadachi magnitnoi gidrodinamiki”, Differents. ur-ya, 27:5 (1991), 875–886  mathnet  mathscinet  zmath
37. V. N. Samokhin, “O sisteme uravnenii magnitnoi gidrodinamiki nelineino vyazkikh sred”, Differents. ur-ya, 27:5 (1991), 886–896  mathnet  mathscinet  zmath
38. V. N. Samokhin, “Suschestvovanie resheniya odnoi modifikatsii sistemy uravnenii magnitnoi gidrodinamiki”, Matem. sb., 182:3 (1991), 395–407  mathnet  mathscinet  zmath
39. M. D. Gunzburger, A. J. Meir, J. S. Peterson, “On the existence, uniqueness, and finite element approximation of solution of the equations of stationary, incompressible magneto-hydrodynamics”, Math. Comp., 56:194 (1991), 523–563  crossref  mathscinet  zmath  adsnasa  isi
40. O. Besson, J. Bourgeois, P.-A. Chevalier, J. Rappaz, R. Touzani, “Numerical model of electromagnetic casting processes”, J. Comput. Phys., 92 (1991), 482–507  crossref  mathscinet  zmath  adsnasa  isi
41. J. Rappaz, R. Touzani, “Modelling of a two-dimensional magnetohydrodynamic problem”, Eur. J. Mech. B/Fluids, 10:5 (1991), 451–453  mathscinet
42. J. Rappaz, R. Touzani, “On a two-dimensional magnetohydrodynamic problem. 1. Modelling and Analysis”, Rairo Modél. Math. Anal. Numer, 26:2 (1992), 347–364  mathscinet  zmath
43. V. N. Samokhin, “O statsionarnykh zadachakh magnitnoi gidrodinamiki nenyutonovskikh sred”, Sib. matem. zhurn., 33:4 (1992), 120–127  mathnet  mathscinet
44. M. Spada, H. Wobig, “On the existence and uniqueness of dissipative plasma equilibria in a toroidal”, J. Phys. A., 25 (1992), 1575–1591  crossref  mathscinet  zmath  adsnasa  isi
45. G. Ströhmer, “About an initial-boundary value problem from magneto-hydrodynamics”, Math. Z., 209 (1992), 345–362  crossref  mathscinet  zmath  isi
46. A. J. Meir, “The equations of stationary, incompressible magnetohydrodynamics with mixed boundary conditions”, Comp. Math. Applic., 25 (1993), 13–29  crossref  mathscinet  zmath
47. A. J. Meir, P. G. Schmidt, “A velocity-current formulation for stationary MHD flow”, Appl. Math. Comp., 65 (1994), 95–109  crossref  mathscinet  zmath
48. L. S. Hou, A. J. Meir, “Boundary optimal control of MHD flows”, Appl. Math. Optim., 32 (1995), 143–162  crossref  mathscinet  zmath  isi
49. G. Milone, V. A. Solonnikov, “On an initial boundary-value problem for equations of magnetohydrodynamics with Hall and ion-sleep effect”, Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 221, 1995, 167–184  mathnet
50. G. Milone, V. A. Solonnikov, “On the solvability of some initial boundary-value problems of magnetofluidmechanics with Hall and ion-sleep effects”, Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Mat. Natur., 6 (1995), 117–132  mathscinet
51. M. P. Galanin, Yu. P. Popov, Kvazistatsionarnye elektromagnitnye polya v neodnorodnykh sredakh (matematicheskoe modelirovanie), Nauka, Fizmatlit, M., 1995  mathscinet  zmath
52. L. Hou, S. Ravindran, “Computations of boundary optimal control problems for an electrically conducting fluid”, J. Comput. Phys., 128 (1996), 319–330  crossref  mathscinet  zmath  adsnasa  isi
53. A. J. Meir, P. G. Schmidt, “Variational methods for stationary MHD flow under natural interface conditions”, Nonlinear Analysis, 26:4 (1996), 659–689  crossref  mathscinet  zmath  isi
54. J.-F. Gerbeau, C. Le Bris, “Existence of solution for a density-dependent magnetohydro- dynamic equation”, Adv. Differential Equations, 2:3 (1997), 427–452  mathscinet  zmath
55. J.-F. Gerbeau, C. Le Bris, “On a coupled system arising in magnetohydrodynamics”, Appl. Math. Lett., 12 (1999), 53–57  crossref  mathscinet  zmath
56. M. Wiedmer, “Finite element approximation for equations of magnetohydrodynamics”, Math. Comp., 69:229 (1999), 83–101  crossref  mathscinet  adsnasa  isi
57. A. J. Meir, P. G. Schmidt, “Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary”, SIAM J. Numer. Anal., 36 (2000), 1304–1332  crossref  mathscinet  isi
58. J.-F. Gerbeau, “A stabilized finite element method for the incompressible magnetohydro-dynamics equations”, Numer. Math., 87 (2000), 83–111  crossref  mathscinet  zmath  isi
59. D. Schötzau, “Mixed finite element methods for stationary incompressible magnetohydro-dynamics”, Numer. Math., 96 (2004), 771–880  crossref  mathscinet  isi
60. V. Girault, P. A. Raviart, “Finite element methods for Navier – Stokes equations”, Theory and algorithms, Springer-Verlag, Berlin, 1986, 376 pp.  mathscinet  zmath
61. A. Valli, Orthogonal decompositions of $L^2(\Omega)^3$, Preprint UTM 493., Department of Mathematics. University of Trento, 1995
62. A. Alonso, A. Valli, “Some remarks on the characterization of the space of tangential traces of $H(rot; \Omega)$ and the construction of the extension operator”, Manuscr. Math., 89 (1996), 159–178  crossref  mathscinet  isi
63. M. Cessenat, “Mathematical methods in electromagnetism”, Linear theory and applications, 41 (1996), Word Scientific Publishing  mathscinet
64. A. Buffa, M. Costabel, D. Sheen, “On traces for $\mathbf H(\mathbf{curl},\Omega)$ in Lipschitz domains”, J. Math. Anal. Appl., 276:2 (2002), 845–876  crossref  mathscinet  isi
65. A. Alonso, A. Valli, “An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations”, Math. Comp., 68 (1999), 607–631  crossref  mathscinet  zmath  adsnasa  isi
66. J. Saranen, “Degeneralized harmonic fields in domain with anisotropic nonhomogeneous media”, J. Math. Anal. Appl., 88 (1982), 104–115  crossref  mathscinet  zmath  isi
67. J. Saranen, “On electric and magnetic problems for vector fields in anisotropic nonhomog-eneous media”, J. Math. Anal. Appl., 91 (1983), 254–275  crossref  mathscinet  zmath  isi
68. M. E. Bogovskii, “Reshenie nekotorykh zadach vektornogo analiza, svyazannykh s operatorami div i grad”, Teoriya kubaturnykh formul i prilozheniya funktsionalnogo analiza k zadacham matematicheskoi fiziki, IM SO AN SSSR, Novosibirsk, 1980, 5–40  mathscinet
69. R. Picard, “On the boundary value problems of electro- and magnetostatics”, Proc. Royal Soc. Edinburgh., 92A (1982), 165–174  mathscinet  zmath
70. P. R. Kotiuga, P. P. Silvester, “Vector potential formulation for three-dimensional magnetostatics”, J. Appl. Phys., 53 (1982), 8399–8401  crossref  adsnasa  isi
71. J. Bolik, W. von Wahl, “Estimating $\nabla{\mathbf u}$ in terms of $\mathrm{div}{\mathbf u}$, $\mathrm{curl}{\mathbf u}$ and either ${\mathbf u}\cdot\nu$ or ${\mathbf u}\times\nu$ and the topology”, Math. Meth. Appl. Sci., 20 (1997), 737–744  crossref  mathscinet  zmath  isi
72. D. Mitrea, M. Mitrea, J. Pipher, “Vector potential thory on nonsmooth domains in $R^3$ and applications to electromagnetic scattering”, J. Fourier Anal. Appl., 3:2 (1997), 131–192  crossref  mathscinet  zmath  isi
73. P. Fernandes, G. Gilardi, “Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions”, Math. Mod. Meth. Appl. Sci., 7 (1997), 957–991  crossref  mathscinet  zmath
74. G. Auchmuty, “Reconstruction of the velocity in the three-demensional fluid flows”, Proc. Royal. Soc. Lond., 454A (1998), 607–630  crossref  mathscinet  zmath  adsnasa
75. C. Amrouche, C. Bernardi, M. Dauge, V. Girault, “Vector potentials in three-dimensional non-smooth domains”, Math. Meth. Appl. Sci., 21 (1998), 823–864  crossref  mathscinet  zmath  isi
76. G. Auchmuty, J. C. Alexander, “$L^2$-well-posedness of planar div-curl systems”, Arch. Rat. Mech. Anal., 160 (2001), 91–134  crossref  mathscinet  zmath
77. G. Auchmuty, J. Alexander, $L^2$-well posedeness of div-rot systems in space, Preprint, 2002
78. G. V. Alekseev, Teoreticheskii analiz obratnykh ekstremalnykh zadach dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi neszhimaemoi zhidkosti, Preprint № 1 IPM DVO RAN, Dalnauka, Vladivostok, 2002, 78 pp.
79. G. V. Alekseev, “Razreshimost statsionarnykh zadach granichnogo upravleniya dlya uravnenii teplovoi konvektsii”, Sib. mat. zhurn., 39:5 (1998), 982–998  mathnet  mathscinet  zmath
80. G. V. Alekseev, “Statsionarnye zadachi granichnogo upravleniya dlya uravnenii teplovoi konvektsii”, Dokl. RAN, 362:2 (1998), 174–177  mathnet  mathscinet  zmath
81. G. V. Alekseev, A. B. Smyshlyaev, D. A. Tereshko, Neodnorodnye kraevye zadachi dlya statsionarnykh uravnenii teplomassoperenosa, Preprint № 19 IPM DVO RAN, Dalnauka, Vladivostok, 2000, 60 pp.
82. G. V. Alekseev, “Obratnye ekstremalnye zadachi dlya statsionarnykh uravnenii teplomassoperenosa”, Dokl. RAN, 375:3 (2000), 315–319  mathnet  mathscinet
83. G. V. Alekseev, E. A. Adomavichus, “Theoretical analysis of inverse extremal problems of admixture diffusion in viscous fluids”, J. Inv. Ill-Posed Probl., 9 (2001), 435–468  mathscinet  zmath
84. G. V. Alekseev, “Razreshimost obratnykh ekstremalnykh zadach dlya statsionarnykh uravnenii teplomassoperenosa”, Sib. mat. zhurn., 42:5 (2001), 971–991  mathnet  mathscinet  zmath
85. G. V. Alekseev, E. A. Adomavichyus, “Issledovanie obratnykh ekstremalnykh zadach dlya nelineinykh statsionarnykh uravnenii perenosa veschestva”, Dalnevost. matem. zhurn., 3:1 (2002), 79–92  mathnet
86. G. V. Alekseev, “Obratnye ekstremalnye zadachi dlya statsionarnykh uravnenii teorii massoperenosa”, Zh. vychisl. mat. i mat. fiz., 42:3 (2002), 380–394  mathnet  mathscinet  zmath
87. A. D. Ioffe, V. M. Tikhomirov, Teoriya ekstremalnykh zadach, Nauka, M., 1974  mathscinet  zmath
88. G. V. Alekseev, R. V. Brizitskii, “Razreshimost smeshannoi zadachi dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi zhidkosti”, Dalnevost. mat. zhurn., 3:2 (2002), 285–301  mathnet  mathscinet
89. G. V. Alekseev, R. V. Brizitskii, “O razreshimosti smeshannoi kraevoi zadachi dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi neszhimaemoi zhidkosti”, Vychisl. tekhnol., 7:1, spets. vyp. (2002), 242–250  zmath
90. G. V. Alekseev, R. V. Brizitskii, “Razreshimost obratnykh ekstremalnykh zadach dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi zhidkosti so smeshannymi granichnymi usloviyami”, Dalnevost. mat. zhurn., 4:1 (2003), 108–126  mathnet
91. G. V. Alekseev, “Zadachi upravleniya dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi neszhimaemoi zhidkosti”, Prikl. mekh. tekhn. fiz., 44:6 (2003), 170–179  mathscinet  zmath
92. G. V. Alekseev, “Zadachi upravleniya dlya statsionarnykh uravnenii magnitnoi gidrodinamiki”, Dokl. RAN, 395:3 (2004), 322–325  mathnet  mathscinet
93. G. V. Alekseev, “Razreshimost zadach upravleniya dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi zhidkosti”, Sib. mat. zhurn., 45:2 (2004), 243–262  mathnet  mathscinet
94. G. V. Alekseev, “O edinstvennosti resheniya zadachi upravleniya dlya statsionarnoi modeli magnitnoi gidrodinamiki vyazkoi neszhimaemoi zhidkosti”, Dalnevost. mat. zhurn., 5:1 (2004), 142–157  mathnet
95. G. V. Alekseev, Teoreticheskii analiz zadach upravleniya dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi teploprovodnoi zhidkosti, Preprint № 1 IPM DVO RAN, Dalnauka, Vladivostok, 2004, 80 pp.
96. G. V. Alekseev, “Obratnye ekstremalnye zadachi dlya statsionarnoi modeli magnitnoi gidrodinamiki teploprovodnoi zhidkosti”, Vychisl. tekhnol., 9:1, spets. vyp. (2004), 158–166
97. R. V. Brizitskii, “Zadachi upravleniya dlya modeli MGD vyazkoi teploprovodnoi zhidkosti so smeshannymi granichnymi usloviyami”, Dalnevost. mat. zhurn., 5:2 (2004), 226–238  mathnet
98. G. V. Alekseev, “Kraevye zadachi i zadachi upravleniya dlya statsionarnoi modeli magnitnoi gidrodinamiki vyazkoi teploprovodnoi zhidkosti”, Dokl. RAN, 405:6 (2005), 744–748  mathnet  mathscinet
99. G. V. Alekseev, R. V. Brizitskii, “Zadachi upravleniya dlya statsionarnykh uravnenii magnitnoi gidrodinamiki vyazkoi teploprovodnoi zhidkosti so smeshannymi granichnymi usloviyami”, Zh. vychisl. mat. i mat. fiz., 45:12 (2005), 2131–2147  mathnet  mathscinet  zmath


© Steklov Math. Inst. of RAS, 2025